where \(\hat{H}(k,\tau _{b},N_{b})=\sum _{j=1}^{N_{b}}A_{\tau _{b}}^{j}e^{kA_{\tau _{b}}^{j}}\) 
and \(\mathsf {\mathfrak {\textrm {err}}}[\hat{H}(k,\tau _{b},N_{b})]\) 
is computed as \(\mathsf {\mathfrak {\textrm {err}}}[\hat{G}(k,\tau _{b},N_{b})]\) .
This formula is obtained assuming that \(\hat{H}(k,\tau _{b},N_{b})\) 
and \(\hat{G}(k,\tau _{b},N_{b})\)  are independent [1]}.
The error on \(\hat{I}(\hat{a}(k,\tau _{b},N_{b}),\tau _{b},N_{b})\) 
can then be estimated as
\(\mathsf {\mathfrak {\textrm {err}}}[\hat{I}(\hat{a}(k,\tau _{b},N_{b}),\tau _{b},N_{b})]=\sqrt{k\mathsf {\mathfrak {^{2}\textrm {err}}}[\hat{a}(k,\tau _{b},N_{b})+\mathsf {\mathfrak {\textrm {err}}}[\hat{\lambda }(k,\tau _{b},N_{b})}.\) 
